THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

3.5. Appendix of Chapter 3 53where [x] denotes the integer part of the real x. Denote by B 1 ,. . . ,B N1 (λ,∆) such hyperrectanglesthat cover ∆ and choose λ = | log h| −1/2 , well defined for h < 1. We have, forb ≥ 0,[]P sup |X t | ≥ bt∈∆≤N 1 (λ,∆)∑j=1P[sup |X t | ≥ bt∈B j]N 1 (λ,∆)∑≤ 2 Pj=1[sup X t ≥ bt∈B j]. (3.14)Let j ∈ {1, . . . , N 1 (λ, ∆)}. Using Corollary 14.2 of Lifshits (1995) (cf Annex .1), we obtainfor b ≥ 4 √ 2D(B j , σ j /2)[ ] (P sup X t ≥ b ≤ exp − 1 (b − 4 √ ) ) 22D(Bt∈B j2σj2 j , σ j /2) , (3.15)where σ 2 j = sup t∈Bj E(X 2 t ),D(B j , σ j /2) =∫ σ/20(log NBj (u) ) 1/2du,where N Bj (u) is the minimal number of ρ-balls of radius u necessary to cover B j and ρ isthe semi-metric defined byρ(s, t) = ( E [ (X s − X t ) 2]) 1/2, s, t ∈ ∆,where E is the expectation with respect to P. Let us evaluate σj 2 . We have, by a changeof variables,(σj 2 1= supQt∈B jh 1 · · · h d∫∆2 u1 − t 1, . . . , u )d − t ddu ≤ ‖Q‖ 2h 1 h2. (3.16)d ∣ Let s, t ∈ B j . For h small enough, we have ∣ s i−t i ∣∣h i< 1 and, using (3.12) and a change ofvariables, we obtaind∑∣ ρ(s, t) ≤ c 1s i − t i ∣∣∣α i∣ . (3.17)h iIn view of (3.17), we have a rough bound for h small enough( [d∏ (λ ]) 1/αiN Bj (u) ≤ N 1 (u, B j ) ≤ 1 +.u)Thus for h small enough4 √ 2D(B j , σ j /2) ≤ 4 √ 2≤ 4λ √ 2∫ λ0d∑i=1i=1i=1[log (N 1 (u, B j ))] 1/2 du ≤ 4λ √ 2∫ 10[log(1 + u−1/α i)] 1/2du.∫ 10[ d∑i=1log ( 1 + u −1/α i )] 1/2du